The Constructions for Large Sets and Overlarge Sets of Resolvable Hybrid Triple Systems

Abstract:

      An LRHTS$(v)$~(or LARHTS$(v))$ is a collection of $\{(X , {\cal B }_i):1\leq i \leq 4(v-2)\}$, where $X$ is a $v$-set, each $(X, {\cal B}_i)$ is a resolvable $($or almost resolvable$)$ HTS$(v)$, and all ${\cal B}_i$s form a partition of all cycle triples and transitive triples on $X$. An OLRHTS$(v)~ ($or OLARHTS$(v))$ is a collection $\{(Y\backslash \{y\}, \A_y^j) : y\in Y, j=0, 1, 2, 3\}, $ where $Y$ is a $(v+1)$-set, each $(Y\backslash \{y\}, {\cal A}_y^j)$ is a resolvable $($or almost resolvable$)$ HTS$(v)$, and all ${\cal A}_y^j$s form a partition of all cycle and transitive triples on $Y$. In this paper, we establish some directed and recursive constructions for LRHTS$(v)$, LARHTS$(v)$, OLRHTS$(v)$, OLARHTS$(v)$ and give some new results.